Elastic Sectors Research Articles

Hydride induced embrittlement and fracture of non-hardening metals under hydrogen chemical equilibrium

The stress field in the hydride precipitation zone is examined, under conditions of hydrogen chemical equilibrium and constant temperature, in the case of non-hardening metals, by applying slip-line theory. It is proven that the hydride precipitation zone, in any geometry, is a constant stress area. In this area, the principal stresses are equal to the respective principal stresses, before hydride precipitation, minus the difference of hydrostatic stress before and after hydride precipitation. The general relations are applied to the case of a stationary sharp mode-I plane-strain crack and the deviations from Prandtl-field are derived, in the [-π/4, +π/4] sector ahead of the tip, where hydrides precipitate. In this case, the hydride precipitation sector is characterized by a constant hydride volume fraction. In addition, hydride precipitation is associated with the development of elastic sectors along the crack faces and the reduction of the centered fan sectors; the relation between hydride precipitation zone stress trace and the extent of the centered fan sector is presented. The mode-I plane-strain blunted crack is also considered and the deviations from the logarithmic spiral slip-lines is discussed together with the reduction of hydride volume fraction as the blunted crack-tip is approached. A general fracture criterion, based on the strength of hydride platelets, is derived, which indicates that fracture occurs, when a critical hydride precipitation zone stress trace dominates. The criterion is applied, under the condition of a dominant K-field annulus, surrounding the plastic zone, and the estimated threshold stress intensity factor of delayed hydride cracking correlates favorably with experimental measurements.

Effect of constraint and latent hardening ratio on the plastic flow around a crack tip in a hardening FCC single crystal

In this work, the crack tip fields in the neighborhood of a stationary mode I crack in a hardening Cu single crystal, under tensile load was investigated. A modified boundary layer (MBL) simulation was performed with the implementation of Bassani and Wu (1991) hardening model while taking into account the crystal elastic anisotropy. The effect of latent hardening ratio (LHR) on the stress and strain fields was studied by considering three different values of LHR, q, viz. diagonal hardening (q=0) and latent hardening (q=1,1.4). In addition, the effect of crack tip constraint representing different test specimens and load conditions was examined with the inclusion of T-stress through the imposed displacement boundary condition. Important observations from this work include the suppression of kink band, presence of elastic sectors, and the occurrence of triple slip near the crack tip. The increase of T-stress suppressed the plastic strain accumulation and showed a decrease in the number of triple slip sectors near the crack tip. Whereas, the increase in q made the triple slip regions slightly prominent with wider angular spread for non-negative T. But, for negative T case, rise in q caused a suppression in the angular spread of triple slip regions. Also, the implementation of T-stress allowed for a correlation between full 3D and 2D plane strain simulations in the analysis of crack tip fields for hardening single crystals. The present study offers a comprehensive investigation of the effect of crack tip constraint and latent hardening ratio on the crack tip fields for hardening FCC single crystals.

The orders of singular stresses at plane bi‐ and trimaterial junctions ‐ Closed‐form analytical solutions

AbstractThe plane problem of a bi‐ or trimaterial‐junction, consisting of dissimilar, homogeneous, isotropic and linear elastic sectors is considered. The asymptotic behaviour of the stresses of this composite situation is analyzed by the complex variable method, based on an appropriate choice of the Kolosov‐potentials which are applicable in the vicinity of the vertex. In the analyses, the identification of the singularity exponent is emphasized. With the help of a novel approach it is demonstrated how to derive some solutions for the orders of the stress singularities at bi‐ and trimaterial combinations in a closed‐form analytical manner. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Wedge indentation into elastic–plastic single crystals. 2: Simulations for face-centered cubic crystals

The asymptotic stress and deformation fields associated with the contact point singularity of a nearly-flat wedge indenter impinging on a specially-oriented single face-centered cubic crystal are derived analytically in a companion paper. In order to investigate the extent of the asymptotic fields, the indentation process is simulated numerically using single crystal plasticity. The quasistatically translating asymptotic fields consist of four angular elastic sectors separated by plastically deforming sector boundaries, as predicted in the companion paper. The asymptotic stress distributions are in accord with the analytical predictions. In addition, simulations are performed for a wedge indenter with a 90° included angle in order to investigate the consequences of finite deformation and finite lattice rotation. Several salient features of the deformation field for the nearly-flat indenter persist in the deformation field for the 90° wedge indenter. The existence of the salient features is validated by comparison to experimental measurements of the lower bound on geometrically necessary dislocation (GND) densities.

Elastic perfectly-plastic asymptotic mixed mode crack tip fields in plane stress

Elastic perfectly-plastic asymptotic plane stress crack tip fields have been constructed by assembling elastic, constant stress and fan sectors under a complete range of mixed mode I/II states of loading. The angular stress distributions are fully continuous, and do not contain the stress discontinuities which have been a feature of many previously proposed solutions. The analytic solutions are verified by finite element solutions under contained yielding conditions. The structure of the elastic perfectly-plastic fields is compared to the structure of the asymptotic strain hardening fields.

Asymptotic crack tip stress, rotation pseudo-stress and dislocation fields for mixed mode I and II cracks in elastic perfectly plastic solids

Nemat-Nasser and Obata (Mech. Mater. 3 (1984) 235) have pointed out that asymptotic stress fields can be constructed around the tip of a general mixed mode I and II crack in elastic perfectly plastic solids with use of elastic sectors whose stress magnitude is below the yield stress. When elastic sectors which are below the yield stress exist the rotation pseudo-stress within an elastic sector is logarithmic singular with distance r from the crack tip. It must also be logarithmic singular within a plastic fan sector that is adjacent to an elastic sector. The areal dislocation density field within a plastic fan adjacent is, therefore, singular as ( r 0/ r)ln( r 0/ r). To justify the existence of below yield stress elastic sectors it is necessary to show that the logarithmic singular stress field produced by the crack plane surface dislocation distribution is canceled by another logarithmic singular stress field produced by the areal dislocation distributions within the fan plastic sectors. This is done so explicitly in this paper.

A family of plane strain crack tip stress fields for interface cracks in strength-mismatched elastic–perfectly plastic solids

Asymptotic crack tip stress fields are developed for a plane strain crack located on the interface between two incompressible, elastic–perfectly plastic solids, with matching elastic properties, but mismatched yield strengths. The complete range of normal versus shear tractions on the interface at the crack tip is analyzed. In the limit as the yield strength mismatch vanishes the asymptotic stress fields for the interface crack approach established crack tip fields for homogeneous bodies. The crack tip stress fields are assembled from combinations of centered fans, constant stress sectors and elastic sectors. All the stresses within each material region are continuous while traction continuity is maintained across the interface. These asymptotic stress fields are verified by comparison with small-scale yielding finite element solutions obtained for various remote load mixity using boundary layer formulations ( Rice, 1967) .

Mode i and mixed mode fields with incomplete crack tip plasticity

Plane strain slip line fields, in which plasticity does not fully surround the crack tip have been developed for mode I and mixed mode I\\II cracks under contained yielding. Analytical solutions have been assembled using slip line theory for the plastic sectors and semi-infinite wedge solutions for the elastic sectors. These solutions are compared with finite element solutions based on modified boundary layer formulations. The analytical solutions agree well with numerical solutions, and form a family of fields with incomplete plasticity around the crack tip.

Plane strain near-tip fields for elastic-plastic interface cracks

The plane strain problem of a stationary interface crack between two dissimilar ductile solids is studied asymptotically, where the ductile solids are assumed to be incompressible, elastic perfectly plastic, and obey the J2-flow theory of plasticity. Candidate asymptotic crack-tip assemblies of plastic and elastic sectors are proposed, and all associated admissible near-tip fields are presented. It is found that when the crack tip is fully surrounded by plastic sectors, then only isolated, mode 1 type solutions exist. When an elastic sector appears along the crack flank in one solid and all other sectors in the two solids are plastic, a two-parameter family of solutions exists, which produces crack-tip stress variations similar to those of the mixed-mode as well as mode I slip-line fields for homogeneous ductile materials. When each of the two solids contains an elastic sector along the crack flank, the crack-tip solutions are found to belong to a four-parameter family, which also resembles mixed-mode and mode I solutions for homogeneous solids. For completeness, the special case of ductile/rigid interfaces is also studied, and several one-parameter families of crack-tip solutions are obtained, which are complementary to those already published in the literature.

Effects of non-singular stresses on crack-tip fields for pressure-sensitive materials, Part 1: Plane strain case

Mode I near-tip stress fields for elastic perfectly plastic pressure-sensitive materials under plane strain and small-scale yielding conditions are presented. A Coulomb-type yield criterion described by a linear combination of the effective stress and the hydrostatic stress is adopted in the analysis. The finite element computational results sampled at the distance of a few crack opening displacements from the tip show that, as the pressure sensitivity increases, the magnitudes of the normalized radial and hoop stress ahead of the tip decrease, the total angular span of the singular plastic sectors decreases, and the angular span of the elastic sectors bordering the crack surfaces increases. When non-singular T stresses are considered along the boundary layer of the small-scale yielding model, the near-tip stresses decrease as the T stress decreases. The plastic zone shifts toward the crack surfaces as the T stress increases. When the discontinuities of the radial stress and the out-of-plane normal stress along the border between the plastic sector and the elastic sector are allowed, the angular variations of the asymptotic crack-tip fields agree well with those of the finite element computations. Variation of the Q stresses for pressure-sensitive materials can be found from the asymptotic solutions when the plastic zone size ahead of the tip is relatively larger than the crack opening displacement. In addition the T stress is shown to have strong effects on the plastic zone sizes and shapes which could affect the toughening of pressure-sensitive materials.

Finite element study of an interface crack between an elastic-perfectly plastic solid and a rigid substrate

The near-tip asymptotic stress fields around an interface crack between an elastic-perfectly plastic solid and a rigid substrate are members of a family parameterized by ζ0, which depends on the phase angle and magnitude of the complex stress intensity factor and bimaterial properties. A new family of three-sector slip-line solutions is found in addition to the existing two-sector field and the family of four-sector fields. The finite element study confirms the existence of this new family, the two-sector solution, and only one member in the four-sector family. The finite element analysis also confirms the existence of elastic sectors near the interfacial crack tip.

Tensile crack-tip fields in elastic-ideally plastic hexagonal crystals and layered materials

Crack tip stress and deformation fields for tensile loaded ideally plastic crystals, as described by Rice [ Mech. Mater. 6, 317 (1987)], have been applied to hexagonal crystals and hexagonal layered solids. The specific case of ( 01 10 ) cracks growing in the [ 2 110 ] direction is considered for hexagonal layered solids, like Pyrolytic graphite (PG) and Carbon ribbon to understand why certain crack orientation and growth directions show unusual fracture toughness values. For the growing crack case, despite the same crack orientation in the two materials, two different types of stress fields are observed, as the orientation of the weakest slip planes with respect to the crack plane is different for the two cases. Furthermore, only one type of elastic sector is present for PG. This is in contrast to the cases of cracks growing in f.c.c. and b.c.c. crystals, wherein two kinds of elastic sectors were present. Surprisingly for (0001) cracks growing in the ( 2 110 ) direction in hcp crystals of Zn and Ti no kinematically permissible solutions were found.

Effects of elasticity and pressure-sensitive yielding on plane-stress crack-tip fields

The asymptotic plane-stress mode I crack-tip fields under small-scale yielding for pressure-sensitive materials are investigated. The yield criterion for these materials is described by a linear combination of the effective stress and the hydrostatic stress. Plastic dilatancy is introduced by normality flow rule. A closed-form general asymptotic solution for singular centered fan sectors is given as a function of μ, which is a pressure sensitivity parameter introduced in the yield condition. When elastic-perfectly plastic behavior is considered, the finite element results show the existence of elastic sectors bordering the stress-free crack faces. The near-tip stresses of the finite element results agree well with those of the corresponding asymptotic analysis. The angular spans of the elastic and plastic sectors vary with the value of μ. The parameter μ also has significant effects on the sizes and shapes of the plastic zones. The contribution of hydrostatic stress in the yield criterion for this class of pressure-sensitive materials extends the boundary of the plastic zone much farther in front of the crack-tip than that for incompressible Mises materials.

Elastic-plastic analysis of cracks in pressure-sensitive materials

In this paper we present an elastic-plastic analysis of the small-scale yielding crack-tip fields for pressure-sensitive materials. Mode I loading and plane-strain conditions are assumed. The yield criterion is chosen to be a linear combination of the effective stress and the hydrostatic stress. For power-law hardening materials, our elastic-plastic finite element analysis shows that HRR-type crack-tipe fields are obtained not only for μ ⩽ μlim, but also for a range of μ >μhm. Here μlim is referred to as the limit value of a pressure sensitivity parameter μ in Li and Pan [J. Appl. Mech. 57, 40–49 (1990)]. When elastic-perfectly plastic behavior is considered, the finite element results show that elastic sectors exist near the crack tip. The sizes of the elastic sectors vary with μ. Plastic zones are also given for different values of the pressure sensitivity parameter μ and the hardening exponent n. The parameter μ has a significant effect on the plastic zone sizes and shapes. The contribution of the hydrostatic stress in the yield criterion causes the plastic zone boundary in front of the crack tip to extend much farther than that in an incompressible material.

Plane-stress mixed-mode near-tip fields in elastic perfectly plastic solids

Within the context of the small-strain approach, the plane-stress mixed-mode near-tip fields of a stationary crack in an elastic perfectly plastic solid under small-scale yielding conditions are examined by finite element methods. The finite element results show that asymptotically at the crack tip two elastic sectors exist under near mode I mixed-mode loading conditions, and one elastic sector exists under near mode II mixed-mode loading conditions. The fully yielded near-tip field, plastically deformed at all angles, is obtained only under pure mode II loading conditions. The corresponding asymptotic crack-tip solutions (consisting of constant stress sectors, curved fan sectors, and elastic sectors) are also constructed. The asymptotic crack-tip stress solutions agree well with the finite element results for the complete range of mixed-mode loadings. Some similarities and differences between the near-tip fields under plane-stress and plane-strain conditions are also discussed.

Tensile crack tip fields in elastic-ideally plastic crystals

Crack tip stress and deformation fields are analyzed for tensile-loaded ideally plastic crystals. The specific cases of (0 1 0) cracks growing in the [1 0 1] direction, and (1 0 1) cracks in the [0, 1, 0] direction, are considered for both fcc and bcc crystals which flow according to the critical resolved shear stress criterion. Stationary and quasistatically growing crack fields are considered. The analysis is asymptotic in character; complete elastic-plastic solutions have not been determined. The near-tip stress state is shown to be locally constant within angular sectors that are stressed to yield levels at a stationary crack tip, and to change discontinuously from sector to sector. Near tip deformations are not uniquely determined but fields involving shear displacement discontinuities at sector boundaries are required by the derived stress state. For the growing crack both stress and displacement must be fully continuous near the tip. An asymptotic solution is given that involves angular sectors at the tip that elastically unload from, and then reload to, a plastic state. The associated near-tip velocity field then has discontinuities of slip type at borders of the elastic sectors. The rays, emanating from the crack tip, on which discontinuities occur in the two types of solutions are found to lie either parallel or perpendicular to the family of slip plane traces that are stressed to yield levels by the local stresses. In the latter case the mode of concentrated shear along a ray of discontinuity is of kink type. Some consequences of this are discussed in terms of the dislocation generation and motion necessary to allow the flow predicted macroscopically.

Numerical analysis of blunting of a crack tip in a ductile material under small-scale yielding and mixed mode loading

The blunting of the tip of a crack in a ductile material is analysed under the conditions of plane strain, small-scale yielding, and mixed mode loading of Modes I and II. The material is assumed to be an elastic-perfectly plastic solid with Poisson's ratio being 1/2. The stress and strain fields for a sharp crack under mixed mode loading are first determined by means of elastic-plastic finite element analysis. It is shown that only one elastic sector exists around the crack tip, in contrast with the possibility of existence of two elastic sectors as discussed by Gao. The results obtained for a sharp crack are used as the boundary conditions for the subsequent numerical analysis of crack tip blunting under mixed mode loading, based on slip line theory. The characteristic shapes of the blunted crack tip are obtained for a wide range of Mode I and Mode II combinations, and found to resemble the tip of Japanese sword. Also the stress field around the blunted crack tip is determined.

Crack tip blunting in ductile material under mixed mode loading by the slip line theory.

Crack tip blunting in ductile material was analyzed under a small-scale yielding subject to mixed mode loadings of Mode I and Mode Ii. The material was assumed to be an elastic perfectly-plastic solid with Poisson's ratio being 1/2. The slip line field for a sharp crack under mixed mode loading was first determined by means of elastic-plastic finite element analysis. It was shown that only one elastic sector exists in the vicinity of the crack tip differing from the possibility of the existence of two elastic sectors discussed by Gao. The result for the sharp crack obtained was used as the boundary condition for the analysis of crack tip blunting. The characteristic shape of the crack tip blunting was obtained, which is just like the tip of a Japanese sword. Also, the stress field was determined around the blunted crack tip.

On stress field near a stationary crack tip

It is well known that the stress and elastic-plastic deformation fields near a crack tip have important roles in the corresponding fracture process. For elastic-perfectly-plastic solids, different solutions are given in the literature. In this work we examine and compare several of these solutions for Mode I (tension), Mode II (shear), and mixed Modes I and II loading conditions in plane strain. By consideration of the dynamic solution, it is shown that the assumption that the material is yielding all around a crack tip may not be reasonable in all cases. By admitting the existence of some elastic sectors, we obtain continuous stress fields even for mixed Modes I and II.

Mode II dynamic fields near a crack tip growing in an elastic-perfectly-plastic solid

A n asymptotic solution is given for Mode II dynamic fields in the neighborhood of the tip of a steadily advancing crack in an incompressible elastic—perfectly-plastic solid (plane strain). It is shown that, like for Modes I and III (G ao and N emat-N asser, 1983), the complete dynamic solution for Mode II predicts a logarithmic singularity for the strain field, but unlike for those modes which involve no elastic unloading, the pure Mode II solution includes two elastic sectors next to the stress-free crack surfaces. This is in contradiction to the quasi-static solution which predicts a small central plastic zone, followed by two large elastic zones, and then two very small plastic zones adjacent to the stress-free crack faces. The stress field for the complete dynamic solution varies throughout the entire crack tip neighborhood, admitting finite jumps at two shock fronts within the central plastic sector. This dynamic stress field is consistent with that of the stationary crack solution, and indeed reduces to it as the crack growth speed becomes zero.